Odd clique minors in graphs with independence number two

TL;DR

This paper investigates the Odd Hadwiger's Conjecture for graphs with independence number at most two, establishing conditions under which such graphs contain large odd clique minors related to their size and structure.

Contribution

It proves that graphs with independence number two contain large odd clique minors if they have certain large cliques or exclude specific induced subgraphs.

Findings

Graphs with independence number ≤ 2 contain odd clique minors of size roughly half their vertices.

Presence of a large clique (≥ n/4) guarantees an odd clique minor of size ⌈n/2⌉.

Exclusion of certain induced subgraphs also ensures the existence of large odd clique minors.

Abstract

A $K_t$-expansion consists of $t$ vertex-disjoint trees, every two of which are joined by an edge. We call such an expansion odd if its vertices can be two-colored so that the edges of the trees are bichromatic but the edges between trees are monochromatic. A graph contains an odd $K_t$ minor or an odd clique minor of order $t$ if it contains an odd $K_t$-expansion. Gerards and Seymour from 1995 conjectured that every graph $G$ contains an odd $K_{\chi(G)}$ minor, where $\chi(G)$ denotes the chromatic number of $G$. This conjecture is referred to as ``Odd Hadwiger's Conjecture". Let $\alpha(G)$ denote the independence number of a graph $G$. In this paper we investigate the Odd Hadwiger's Conjecture for graphs $G$ with $\alpha(G)\le2$. We first observe that a graph $G$ on $n$ vertices with $\alpha(G)\le2$ contains an odd $K_{\chi(G)}$ minor if and only if $G$ contains an odd clique minor of order $\lceil n/2\rceil$. We then prove that every graph $G$ on $n$ vertices with $\alpha(G)\le 2$ contains an odd clique minor of order $\lceil n/2\rceil$ if $G$ contains a clique of order $n/4$ when $n$ is even and $(n+3)/4$ when $n$ is odd, or $G$ does not contain $H$ as an induced subgraph, where $\alpha(H)\le 2$ and $H$ is an induced subgraph of $K_1 + P_4$, $K_2+(K_1\cup K_3)$, $K_1+(K_1\cup K_4)$, $K_7^-$, $K_7$, or the kite graph.